1. Technical Field
The invention relates to an image processing technique for forming a fractal image by using a sub-self-similar set and particularly for making it possible to generate a sub-self-similar set easily.
2. Description of the Related Art
A self-similar set decided from a group of contraction maps is known well as a typical example of fractal. The self-similar set is a set, which is uniquely decided from contraction maps f1, . . . , fn in a complete metric space. The self-similar set is defined as a compact set A satisfying A=∪i−1nf1(A). The definition of the self-similar set is well known, for example, in Non-Patent Document 1-3.
The self-similar set has such a characteristic that the set itself is formed of a sum of n contracted copies. As self-similar sets, there are sets having various forms such as sets named “Cantor set”, “Koch curve”, “Levy curve”, “Sierpinski Triangle” and “Takagi curve.” Particularly, it is known that a part of the self-similar sets are models suitable for complex forms existing in the natural world. For example, the Hata's tree-like set and the Barnsley's fern will correspond to this type of self-similar sets. More generally, it is known that a complex form can be approximated as a self-similar set when contraction maps are selected wisely. Characteristics of these self-similar fractals have been described in detail, for example, in Non-Patent Document 1-3. Such a self-similar set can be drawn easily by a method such as random iteration algorithm or deterministic algorithm described in detail, for example, in Non-Patent Document 2.
The self-similar set can give a model preferred in terms of representation of a complex form or a natural form as described above. However, the number of contraction maps generally increases if it is attempted to obtain a form approximate to an arbitrary form from the self-similar set. Moreover, the approximation of the form on this occasion is based on a distance called “Hausdorff distance.” Therefore, the distance may be different from a distance visually recognized by a human being. In addition, there is a directionality of giving a complex form “likelihood” as another directionality concerned with use of the self-similar set. This directionality may be thought to be a rather preferred directionality in terms of characteristics of the self-similar set. However, an artificial factor remains in the obtained form because of the limitation of “self-similar”.
On the other hand, a compact set P satisfying F⊂∪i=1nf1(F) with respect to the contraction maps f1, . . . , fn is called “sub-self-similar set.” That is, the sub-self-similar set has such a characteristic that the subset itself is included in a set formed of a sum of n contracted copies. The sub-self-similar set is different from the self-similar set in that the sub-self-similar set cannot be decided uniquely. The sub-self-similar set may include a large number of sets. The self-similar set A itself and a boundary set of the self-similar set A satisfy the definition of the sub-self-similar set. As is obvious from this fact, the sub-self-similar set may be said to have a more general figure than that of a fractal decided from a group of contraction maps. Incidentally, mathematical characteristics of the sub-self-similar set have been described in detail, for example, in Non-Patent Document 3.
Because the sub-self-similar set has a wide set including a self-similar set, it is a matter of course that the sub-self-similar set is thought to be more suitable for representation of the complex form of the self-similar set than the self-similar set. Nothing but construction with sofic systems, for example, as disclosed in Non-Patent Document 6 has been however known as a method for drawing the sub-self-similar set. Although these techniques can provide a general method of constructing a sub-self-similar set, it is impossible to know or designate the form of the obtained sub-self-similar set by analogy. For example, it is impossible to obtain a sub-self-similar set having an intended form.
[Non-Patent Document 1] Hutchinson, “Fractals and self-similarity”, Indiana Univ. Math. J. 30 (1981) 713-747.
[Non-Patent Document 2] Barnsley, “Fractals Everywhere” second edition, Academic Press (1993).
[Non-Patent Document 3] Falconer, “Techniques in Fractal Geometry”, John Wiley & Sons (1997).
[Non-Patent Document 4] Hata, “On the structure of self-similar sets”, Japan J. Appl. Math. 2 (1985) 381-414.
[Non-Patent Document 5] S. Dubuc, A. Elqortobi, “Approximations of fractal sets”, J. comput. Appl. Math. 29 (1990) 79-89.
[Non-Patent Document 6] Bandt, “Self-similar sets 3. Construction with sofic systems”, Mh. Math. 108 (1989) 89-102.
[Non-Patent Document 7] Bandt, Keller, “Self-similar sets 2. A simple approach to the topological structure of fractals”, Math. Nachr. 154 (1991) 27-39.
[Non-Patent Document 8] Kameyama, “Self-similar sets from the topological point of view”, Japan J. Ind. Appl. Math. 10 (1993) 85-95.